Hey everyone, great thread!
I am also in the process of debating between programs, so I thought I might share my very informal list -- It's not all necessarily visit-day questions, but more generally information I think I'll consider in my choice or just generally want to know (some was already mentioned above):
Research & day-to-day:
Regarding potential supervisors:
Experience of other grad students with them
Mentoring style, etc.
Current research projects of faculty & other grad students
Atmosphere (Competitive vs cooperative, etc.) & Social life
Qual/quant/mixed-methods research support & statistical consultation or support
Is it possible/typical to work with more than one faculty
Interdisciplinarity & possibilities for research collaborations
Best areas to live
Money questions:
Duration of funding & modalities of summer funding
Are there fees to be paid aside from the tuition remission, and how much
What's the delay before first pay
Availability of funding for conferences and such
Other:
Graduation/completion rates and average nb of years to PhD
Numbers of student publications / student-faculty publications
Placement records (in terms of TT jobs compared to incoming cohort size, if that's what you're interested in)